Proceedings of the Edinburgh Mathematical Society (Series 2)

Table of Contents - February 2012 - Volume 55, Issue 01  

Research Article

Location of geodesics and isoperimetric inequalities in Denjoy domains

José M. Rodrígueza1 and José M. Sigarretaa2

a1 Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Madrid, Spain (jomaro@math.uc3m.es)

a2 Facultad de Mateméticas, Universidad Autónoma de Guerrero, Carlos E. Adame No. 54 Col. Garita, 39650 Acalpulco, Guerrero, Mexico (josemariasigarretaalmira@yahoo.es)

Abstract

We find approximate solutions (chord–arc curves) for the system of equations of geodesics in Ω∩ℍ for every Denjoy domain Ω, with respect to both the Poincaré and the quasi-hyperbolic metrics. We also prove that these chord–arc curves are uniformly close to the geodesics. As an application of these results, we obtain good estimates for the lengths of simple closed geodesics in any Denjoy domain, and we improve the characterization in a 1999 work by Alvarez et al. on Denjoy domains satisfying the linear isoperimetric inequality.

(Received May 28 2009)

(Online publication June 15 2011)

Keywords

  • geodesic;
  • quasi-geodesic;
  • chord–arc;
  • Poincaré metric;
  • linear isoperimetric inequality;
  • Denjoy domain

2010 Mathematics subject classification

  • Primary 30F45;
  • Secondary 53C22;
  • 30C99